As is well known, the theory of partial differential equations is much more difficult than the theory of ordinary differential equations, because PDEs don't behave as nicely.

I was thinking, however, that the generalization of regular derivatives to the higher-dimensional case is usually best thought of as the total (i.e. Frechet) derivative, rather than partial derivatives.

Question: Is there a theory of total differential equations? If not, why not? And if so, does the theory allow one to sidestep around a lot of the problems which arise in the theory of PDEs?

The main problem I could think of is "type-checking"; namely the original function will be a first-order tensor field, the first derivative would be a second-order tensor field, the second derivative would be a third order tensor field, and so on... Is there not a way to circumvent this problem?

Note: Just so everyone is clear, I am not arguing against the usefulness of the theory of PDEs; on the contrary, they were even used by Perelman to prove the Poincare Conjecture. I'm just wondering, if total differential equations would be easier to study, why no one has studied them.

The existence of partial derivatives is only necessary but not sufficient for the existence of the total derivative. So my conjecture is that solving PDE's restricted to the solution space of total differential equations would be much easier and would have to account for far fewer pathological cases, since the existence of the total derivative implies far more regularity properties than one gets by assuming the existence of the partial derivatives alone. (In fact, the existence of partial derivatives does not even imply the existence of all directional derivatives, which in turn does not even imply Gateaux differentiability, which in turn does not even imply total differentiability, so one should expect the general theory of PDEs to have far more pathological cases than the specific theory of total differential equations.)

In other words, it seems like studying total differential equations might make it easier to identify which classes of PDE are easy to solve and would help explain in part why some PDEs are more difficult to solve, since it would make much more explicitly clear how analogies with ODEs break down in those cases. So even though total differential equations would arguably be a sub-case of the field of PDEs, the additional regularity properties would still make it a worthwhile object of study, the same way that the study of symmetric or diagonalizable matrices illuminates the whole of linear algebra.

  • $\begingroup$ Do you consider the defining equation of harmonic maps a total differential equation? $\endgroup$ – user98602 Aug 18 '16 at 2:56
  • $\begingroup$ @MikeMiller $d\phi$ the differential is the total differential, right? That does look like one then, yes. Thank you for the example which I didn't know about $\endgroup$ – Chill2Macht Aug 18 '16 at 2:58
  • $\begingroup$ Aren't partial derivatives just the components of total derivatives? And then won't equations involving total derivatives just boil down to systems of pde's? $\endgroup$ – Andreas Blass Aug 18 '16 at 3:43
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    $\begingroup$ You may be right; I'm certainly not an expert on regularity results for PDE's. But remember that sometimes the failure of regularity is not mathematical pathology but physical reality. Some physical systems develop shock waves, and mathematical models of them develop corresponding discontinuities. $\endgroup$ – Andreas Blass Aug 18 '16 at 4:07
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    $\begingroup$ Depending on the kind of PDE work you do, you often start by working with distributions of some sort, and then assuming your inputs are in Sobolev spaces. In this situation derivatives always exist by the very nature of interpreting things distributionally; the question is eg if you're $L^2$ and the first $k$ of your derivatives are $L^2$, are more derivatives $L^2$? This in mind, I'm not sure that the total derivative helps very much with regularity questions. $\endgroup$ – user98602 Aug 18 '16 at 15:46

Your notion of "total differential equation" is what mathematicians call a "classical solution" of a partial differential equation. A classical solution of a kth order PDE is a k-times continuously differentiable (hence k-times Frechet differentiable) function satisfying the PDE.

For some PDEs (e.g., linear uniformly elliptic equations), we can prove that classical solutions exist and are unique, and many major questions have been answered.

Unfortunately, for very many very important PDEs, classical solutions simply do not exist (for example, Hamilton-Jacobi equations, degenerate elliptic equations, conservation laws, to name a few). That is to say, you can prove there are no k-times continuously differentiable functions satisfying the PDE. Nevertheless, these PDEs are very important and should have solutions in some sense.

Thus, one is forced by the nature of the problem to consider weaker notions of solution (entropy, viscosity, distributional, etc.). These are notions which allow functions that are not k-times continuously differentiable to solve the PDE in some sense. The point is that this is not a choice; it is imposed on us by the nature of certain PDEs. In fact, some of us would say this is what makes PDEs interesting to study!

  • $\begingroup$ I am a little confused about the first paragraph. What I am trying to ask, for example, is given a function $f(x,y): \mathbb{R}^2 \to \mathbb{R}^2$, we can write equations involving any or all of its partial derivatives $f^1_x, f^2_x, f^1_y$ and $f^2_y$. If the equation involves all four partial derivatives and they are $k-$times continuously differentiable, you are right that we are done. What about when the equation involves only $f^1_x$ and $f^2_y$ for example? If they are continuously differentiable what can we say about the functions $f^1_y$ and $f^2_x$ that are not in the equation? $\endgroup$ – Chill2Macht Sep 3 '16 at 23:57
  • $\begingroup$ Then again though, if we are talking about total differential equations, then all of the partial derivatives have to be in the equation, so the above objection is spurious. Even though continuous differentiability of the partial derivatives is technically only a sufficient but not necessary condition for total differentiability, it is the only condition I am aware of in practice that people use. So your observation that, maybe with minor exceptions, any theory of TDEs is encompassed already in the standard theory of PDEs does seem correct. Thank you for this clear argument. $\endgroup$ – Chill2Macht Sep 4 '16 at 0:01
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    $\begingroup$ @William You are right, normally a PDE does not include all partial derivatives up to a certain order $k$. If you required that it did, then you would be excluding most important PDE from your study. Keep in mind the notion of classical solution requires all partials up to order $k$ to exist and be continuous, even if they do not appear in the equation. So for a classical solution of the heat equation $u_t=u_{xx}$, $u_{xt}$ and $u_{tt}$ would have to exist and be continuous, even though they do not appear in the equation (sometimes we don't require $u_{tt}$, depends on the text). $\endgroup$ – Jeff Sep 4 '16 at 0:19
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    $\begingroup$ @William I should have mentioned in my original answer that the term Frechet differentiability is usually reserved for infinite dimensional spaces (like Banach spaces). In Euclidean space $\mathbb{R}^n$, Frechet differentiability is equivalent to what we usually just call differentiability (see any analysis text, for instance Rudin "Real and Complex Analysis"). $\endgroup$ – Jeff Sep 4 '16 at 0:24
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    $\begingroup$ @William Yes, Evans' book is great, and is the standard graduate text for PDE. You can find a discussion of classical vs weak solutions, and their importance in PDE in the introduction of Evans' book. Generally speaking, you can get through chapters 1-4 in Evans with only some real analysis and multivariable calculus. Beware that to go beyond this (chapters 5-12), you'll need some graduate real analysis and functional analysis. $\endgroup$ – Jeff Sep 4 '16 at 2:09

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